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Related Experiment Video

Updated: May 18, 2026

Design and Use of Multiplexed Chemostat Arrays

Design and Use of Multiplexed Chemostat Arrays

Published on: February 23, 2013

Behavior of a single element in a finite stochastic array.

José Gómez-Ordóñez¹, José M Casado, Manuel Morillo

¹Área de Física Teórica, Facultad de Física, Universidad de Sevilla, Apartado de Correos 1065, Sevilla 41080, Spain.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 26, 2012

Summary

The nonlinear Fokker-Planck equation (NLFPE) reliably models finite stochastic systems with one solution. However, it deviates from simulations for systems with two coexisting solutions.

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Area of Science:

Statistical physics
Nonlinear dynamics
Computational physics

Background:

Mean-field approximations simplify complex many-body systems.
The nonlinear Fokker-Planck equation (NLFPE) models single-element dynamics in infinite systems.
Finite systems may exhibit different behaviors than predicted by infinite-system approximations.

Purpose of the Study:

To assess the accuracy of the NLFPE for finite nonlinear stochastic arrays.
To compare NLFPE predictions with direct simulations of coupled Langevin equations.
To identify conditions where the NLFPE approximation is reliable or breaks down.

Main Methods:

Numerical analysis of the nonlinear Fokker-Planck equation.
Numerical simulations of the full set of Langevin equations for coupled elements.
Comparison of stationary and time-evolution properties between NLFPE and simulations.

Main Results:

The NLFPE accurately describes finite systems with a single stationary solution.
For parameter regimes with two coexisting stationary solutions, the NLFPE significantly diverges from simulation results.
The reliability of the NLFPE is parameter-dependent, particularly concerning the number of stable solutions.

Conclusions:

The NLFPE is a valid approximation for finite stochastic systems under specific conditions (single stationary solution).
The mean-field ansatz underlying the NLFPE fails to capture crucial dynamics in finite systems exhibiting multiple stable states.
Careful validation against direct simulations is essential when applying the NLFPE to finite nonlinear stochastic systems, especially those with complex solution landscapes.